How to calculate the eigenvalues of a matrix?

To calculate the eigenvalues of a matrix, we need to find the values of λ that satisfy the equation det(A-λI) = 0.

First, we need to subtract λ from the diagonal elements of the matrix A and create a new matrix (A-λI). Then, we need to find the determinant of this new matrix using any method we prefer (e.g. expansion by minors, row reduction, etc.). The resulting equation will be a polynomial of degree n, where n is the size of the matrix.

Next, we need to solve this polynomial equation to find the values of λ that satisfy det(A-λI) = 0. These values are the eigenvalues of the matrix A.

For example, let's find the eigenvalues of the matrix A = [2 1; 1 2]:

det(A-λI) = det([2-λ 1; 1 2-λ])
= (2-λ)(2-λ) - 1
= λ^2 - 4λ + 3
= (λ-1)(λ-3)

Therefore, the eigenvalues of A are λ = 1 and λ = 3.

It is important to note that the eigenvalues of a matrix are not always real numbers. In some cases, they may be complex numbers. Additionally, if the matrix is not diagonalizable, it may not have enough eigenvectors to form a basis for the vector space.